Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470894 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#121 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1823 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2,3,4)(7,11)(10,12)(13,14)(15,23)(16,21)(17,19)(18,30)(20,27)(22,24)(25,36)(26,35)(28,32)(29,41)(31,37)(33,40)(34,38)(39,43)(42,44);(2,4)(5,6)(7,10)(8,9)(11,12)(14,19)(20,31)(21,32)(22,30)(23,35)(33,42)(34,41)(36,43);(6,9)(10,12)(13,15,24,37,17,26,18,27)(14,20,30,35,19,31,22,23)(16,25,38,44,28,39,29,40)(21,33,41,43,32,42,34,36);(6,9)(10,12)(13,16)(14,21)(15,25)(17,28)(18,29)(19,32)(20,33)(22,34)(23,36)(24,38)(26,39)(27,40)(30,41)(31,42)(35,43)(37,44);(5,7,8,11)(6,10,9,12)(15,26)(16,28)(20,31)(21,32)(23,35)(27,37)(29,38)(34,41)Polycyclic presentation:
2^9|a^2=fi,c^2=gi,e^2=h,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=dh,e^c=ei,e^d=ei80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6401336);